reserve X for RealNormSpace;

theorem
  for A be non empty closed_interval Subset of REAL,
      f1, f2 be PartFunc of REAL,the carrier of X
        st f1 is_integrable_on A & f2 is_integrable_on A &
        A c= dom f1 & A c= dom f2 holds
        f1 - f2 is_integrable_on A &
        integral(f1 - f2,A) = integral(f1,A) - integral(f2,A)
proof
  let A be non empty closed_interval Subset of REAL;
  let f1, f2 be PartFunc of REAL,the carrier of X;
  assume that
A1: f1 is_integrable_on A &
    f2 is_integrable_on A and
A2: A c= dom f1 & A c= dom f2;
A3: f1 - f2 = f1 + -f2 by VFUNCT_1:25;
A4: dom (-f2) = dom f2 by VFUNCT_1:def 5;
A5: -f2 = (-jj)(#)f2 by VFUNCT_1:23; then
A6: -f2 is_integrable_on A by A1,A2,Th13;
  hence (f1 - f2) is_integrable_on A by A1,A2,A3,A4,Th14;
  thus integral(f1 - f2,A) = integral(f1 + -f2,A) by VFUNCT_1:25
    .= integral(f1,A) + integral(-f2,A) by A1,A2,A4,A6,Th14
    .= integral(f1,A) + (-jj) * integral(f2,A) by A1,A2,A5,Th13
    .= integral(f1,A) + -integral(f2,A) by RLVECT_1:16
    .= integral(f1,A) - integral(f2,A) by RLVECT_1:def 11;
end;
